Solve for $n$, $ \dfrac{8}{5n + 1} = \dfrac{1}{15n + 3} + \dfrac{2n - 7}{25n + 5} $
Answer: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5n + 1$ $15n + 3$ and $25n + 5$ The common denominator is $75n + 15$ To get $75n + 15$ in the denominator of the first term, multiply it by $\frac{15}{15}$ $ \dfrac{8}{5n + 1} \times \dfrac{15}{15} = \dfrac{120}{75n + 15} $ To get $75n + 15$ in the denominator of the second term, multiply it by $\frac{5}{5}$ $ \dfrac{1}{15n + 3} \times \dfrac{5}{5} = \dfrac{5}{75n + 15} $ To get $75n + 15$ in the denominator of the third term, multiply it by $\frac{3}{3}$ $ \dfrac{2n - 7}{25n + 5} \times \dfrac{3}{3} = \dfrac{6n - 21}{75n + 15} $ This give us: $ \dfrac{120}{75n + 15} = \dfrac{5}{75n + 15} + \dfrac{6n - 21}{75n + 15} $ If we multiply both sides of the equation by $75n + 15$ , we get: $ 120 = 5 + 6n - 21$ $ 120 = 6n - 16$ $ 136 = 6n $ $ n = \dfrac{68}{3}$